Always A Happy Voter - Arrow's Impossibility Theorem

I think (08/02/2023) Arrow's Impossibility Theorem is understood as being more dire than it is. The theorem states, no ranking system can be fair, and a non-dictatorship. However, Arrow does not use the colloquial meaning of dictatorship. In my opinion, a better phrasing is, every fair ranking system totally satisfies the preferences of at least one voter.

In Social Choice and Individual Values, Kenneth J. Arrow asks: To what extent can individual preferences be aggregated into a fair societal ordering? This is the same problem as ranked-choice voting, though instead of a winner, an ordering of all options is produced.

Arrow requires three things be true of fair societal orderings.

Transitivity: If $x$ is preferred to $y$ , and $y$ to $z$ , then $x$ must be preferred to $z$ .
Independence of Irrelevant Alternatives: For unique candidates $x$ , $y$ , and $z$ ; if $z$ is removed from the set of options, $x$ 's ranking relative to $y$ does not change.
Unamity: If every individual ranks $x$ over $y$ , the societal ordering must rank $x$ over $y$ .

Surprisingly, Arrow proves any social ordering satisfying these requirements is a dictatorship (in a result now called Arrow's Impossibility Theorem). From page 30 of the second edition, a choice by dictatorship, "In its pure form, it means that social choice are to be based solely on the preferences of one man. That is, whenever the dictator prefers $x$ to $y$ , so does society."

I find Arrow's description misleading. The use of "based solely on the preferences of one man", suggests fair societal orderings can be hijacked; however, in practice, the "dictator" changes across outcomes. What Arrow really proves is, any fair societal ordering completely satisfies the preferences of at least one voter.

Nothing is necessarily wrong with that; the decision process can be perfectly democratic, and one person simply turns out to be on the winning side on all issues.

As Aanund Hylland writes in Foundations of Social Choice Theory, page 51. I recomend Three Brief Proofs of Arrow's Impossibility Theorem by John Geanakoplos for exploring this more formally.